Pronormal implies weakly closed in intermediate nilpotent

Statement

Statement with symbols

Suppose $H\leq K\leq G$ are groups such that:

$H$ is a Pronormal subgroup (?) of $G$ .
$K$ is a Nilpotent group (?).

Then, $H$ is a Weakly closed subgroup (?) of $G$ .

Related facts

Pronormal implies intermediately subnormal-to-normal

Stronger facts

Paranormal implies weakly closed in intermediate nilpotent: Paranormality is a somewhat weaker assumption than pronormality, and hence the corresponding result is stronger. The proof is almost the same.

Definitions used

For these definitions, $H^{g}=g^{-1}Hg$ denotes the conjugate subgroup by $g\in G$ . (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).

Pronormal subgroup

Further information: Pronormal subgroup

A subgroup $H$ of a group $G$ is termed paranormal in $G$ if for any $g\in G$ , there exists $x\in \langle H,H^{g}\rangle$ such that $H^{x}=H^{g}$ .

Weakly closed subgroup

Further information: Weakly closed subgroup

Suppose $H\leq K\leq G$ are groups. We say $H$ is weakly closed in $K$ with respect to $G$ if, for any $g\in G$ such that $H^{g}\leq K$ , we have $H^{g}\leq H$ .

Facts used

Proof

Given: $H\leq K\leq G$ with $H$ a paranormal subgroup of $G$ and $K$ a nilpotent group.

To prove: For any $g\in G$ such that $H^{g}\leq K$ , we have $H^{g}\leq H$ .

Proof: By fact (2), $H$ is a subnormal subgroup of $K$ . By fact (1), $H$ is therefore normal in $K$ .

Now suppose $g\in G$ is such that $H^{g}\leq K$ . Then, $\langle H,H^{g}\rangle \leq K$ . By fact (3), $H$ is normal in $\langle H,H^{g}\rangle$ . Thus, for any $x\in \langle H,H^{g}\rangle$ , we have $H^{x}\leq H$ .

By the definition of pronormality, we also have $x\in \langle H,H^{g}\rangle$ such that $H^{x}=H^{g}\rangle$ . Since $H^{x}\leq H$ , we get $H^{g}\leq H$ , completing the proof.

References

Textbook references

Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 266, Exercise 9, Chapter 7